Information, Bubbles, and the Efficient Markets Hypothesis

Last Updated: 6 March 2012

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Professor Leigh Tesfatsion
tesfatsi AT iastate.edu
Econ 308 Web Site:
https://www2.econ.iastate.edu/classes/econ308/tesfatsion/

The notes, below, provide basic background discussion on information, bubbles, and the efficient markets hypothesis. For a more extensive set of notes relating to these and other financial economics topics prepared for an undergraduate course, visit the home page for Econ 353 (Money, Banking, and Financial Institutions)

  1. Overview
  2. Analytical Representation for the Efficient Markets Hypothesis
  3. Bubbles and Fundamental Values in Asset Pricing: A Simple Illustration
  4. Empirical Testing of the Efficient Markets Hypothesis
  5. References
  6. Appendix: Technical Remarks

A. Overview:

The concept of "efficiency" has a variety of related but distinct meanings in economics. An economy can be efficient in the sense of productive efficiency, meaning non-wastage of physical resources. An economy can also be efficient in the sense of Pareto-efficiency, meaning non-wastage of utility. Finally, an economy can be efficient in the sense of informational efficiency, meaning non-wastage of information.

In finance, the term "efficiency" has come to be used specifically in the last sense. Roughly speaking, a market for financial assets is said to exhibit (informational) efficiency if all available information is optimally used in the determination of asset prices at each point in time. This has been formalized as the so-called "efficient markets hypothesis."

The Efficient Markets Hypothesis (EMH): The contention that, at each time t, the current prices of financial assets reflect all available information relevant for judging the future returns of those assets.

The intuitive idea is that rational individual traders process the information that is available to them and take optimal positions in assets on the basis of this information. The market price for an asset then aggregates this diverse trader information, and in this sense "reflects" the available information.

What is the relationship among these concepts of productive, informational, and Pareto efficiency? Roughly speaking, productive and informational efficiency can be thought of as requirements for an economy to be Pareto efficient. If either physical resources or information are being "wasted," then (essentially by definition of "wasted") there must exist a way to improve the use of resources and/or information to make at least one person better off without hurting anyone else.

B. Analytical Representation for the Efficient Markets Hypothesis

An asset is said to be risky relative to a particular time period if its return rate over this time period is not completely known or assured; otherwise it is said to be risk free. Intuitively, if a market for risky financial assets over period [t,t+1) is to be "efficient," then it must be the case that traders in this market exploit all available profit opportunities, leaving no room for further profits to be made in net terms.

Assuming traders are risk neutral (i.e., they only care about expected return rates, not the volatility of return rates), and that at least one risk-free asset is held, this implies the following arbitrage condition for each risky asset A in this market: At the margin, the return rate that a trader expects to obtain over the period [t,t+1) by investing one last dollar in asset A at time t should just equal the return rate that the trader could obtain by instead investing this dollar in a best (most profitable) available risk-free asset (e.g., U.S. Treasury bills).

REMARK: As noted above, this use of the risk-free return rate as the "opportunity cost" used by traders to judge the desirability of investing in risky financial assets assumes that traders are risk neutral, so that only the expected return rate is of importance to them in comparing the purchase of alternative financial assets. Moreover, which particular asset is to be designated as "the" risk-free asset determining the opportunity cost of traders has also been controversial. Following Fama (1970), the empirical finance literature has come to recognize that EMH tests actually represent joint tests of the EMH, the degree of risk aversion exhibited by traders, and the precise specification of the opportunity cost.

Let the return rate on the best available risk-free asset over period [t,t+1) be denoted by r_F(t,t+1), and let r_A(t,t+1) denote the return rate on some risky asset A over period [t,t+1). Also, let I(t) denote the information available to investors at time t, where I(t) includes the value of r_F(t,t+1). Then the EMH relative to the sequence of information sets {I(t)} states that the expected value of r_A(t,t+1), conditional on I(t), must equal the risk-free return rate r_F(t,t+1). Formally:

(B.1)         E[r_A(t,t+1)| I(t) ]  =   r_F(t,t+1)  ,  t = 0, 1, ... .

Let P_A(t) and P_A(t+1) denote the market prices of asset A at time t and t+1, respectively. Also, let D(t,t+1) denote the amount of payments generated by asset A over the period [t,t+1), assumed to be paid out at time t+1. Then the return rate r_A(t,t+1) for asset A over [t,t+1) is given by

(B.2)         r_A(t,t+1)  =  [P_A(t+1) - P_A(t) + D(t,t+1)]/P_A(t)  .

If the value of P_A(t) is included in the information set I(t), conditions (B.1) and (B.2) together imply that

(B.3)         E[P_A(t+1)+D(t,t+1)|I(t)]  =  [1+r_F(t,t+1)]P_A(t)  ,

or

(B.4)          E[P_A(t+1)+D(t,t+1)|I(t)]
              ---------------------------   =   P_A(t)  .
                   1 + r_F(t,t+1)

Roughly speaking, the EMH implies that the market for information about asset returns is competitive in the following sense. Investors cannot expect to earn persistent monopolistic returns -- i.e., returns in excess of those necessary to induce the investor to hold an asset -- by collecting information about past asset prices and trading volumes and basing their current trades on this information. The reason for this is that the very act of trading in the asset tends to reveal the investor's information. Thus, a dramatic implication of the EMH is that it denies the possibility of successful technical trading schemes. Given the current price of an asset, no clever use of information concerning the past price and trade volume history of an asset can improve one's chances of predicting its future prices.

The fundamental value of an asset is defined to be the present value of its (possibly infinite) payment stream, discounted by the risk-free rate r_F. If the price of an asset deviates from its fundamental value, the asset is said to exhibit a (price) bubble.

Some economists (e.g., Arthur et al. (1997)) analytically represent the EMH in stronger terms than (B.4). They equate the EMH with the absence of bubbles in the pricing of financial assets. That is, they argue that, in a financial market consisting of rational fully-informed traders, the price of each financial asset will equal the asset's fundamental value. In the next section it is shown that the arbitrage condition (B.4) is not sufficient, per se, to rule out the existence of price bubbles.

C. Bubbles and Fundamental Values in Asset Pricing: A Simple Illustration

Consider an economy for which government issued money ("dollars") and government issued debt instruments are the only available financial assets. Suppose the latter take two forms: (1) Risk-free Treasury bills paying a constant risk-free return rate r_F in each period t = [t,t+1); and (2) risky consols, where each consol promises the holder a coupon payment of one dollar at the end of each period in perpetuity (i.e., with no maturity date).

The first question to be addressed concerns the relation of the price of a consol to its return rate. Let P_C(t) denote the nominal price of a consol at the beginning of any period t, i.e., the price measured in dollars. The nominal return from purchasing a consol at time t at price P_C(t), holding the consol for one period, and selling the consol at time t+1 at a price P_C(t+1), is then given by

(C.1)       P_C(t+1) -  P_C(t)             +           $1 per consol

     Return from Price Appreciation                   Coupon Payment
     (Measured in dollars per consol)

Consequently, dropping the "per consol" designations for ease of notation, the nominal return rate attained by holding a consol from time t to time t+1 is given by

                            [P_C(t+1) -  P_C(t) +  $1]
(C.2)       r_C(t,t+1)   =  --------------------------
                                      P_C(t)

Now suppose that the behavior of investors in each period t can be captured by the behavior of a single risk-neutral infinitely-lived "representative" investor whose portfolio contains both Treasury bills earning the constant risk-free return rate r_F and and consols earning the return rate (C.2). For each time t, letting I(t) denote the time-t information set of this investor, suppose the EMH (B.4) holds for consols, implying that

(C.3)    E[r_C(t,t+1)|I(t)]  =  r_F  .

Finally, suppose for each time t that I(t) includes the value of P_C(t), the value of r_F, and the value of the $1 payments promised by each consol, and that successive information sets I(t) and I(t+1) are "nested" in the sense that I(t) is included in I(t+1).

Using (C.2) to substitute out for r_C(t,t+1) in (C.3), and multiplying through the resulting expression by P_C(t) and solving for P_C(t), one obtains the following reformulation of condition (C.3):

                   $1      E[P_C(t+1)|I(t)]
(C.4)  P_C(t) = ------- +  ----------------
                (1+r_F)         (1+r_F)

By assumption, relation (C.4) holds for every time t. Consequently, updating (C.4) by one time period, one gets an expression for P_C(t+1) that can be used to substitute out for the far-right appearance of P_C(t+1) in the original expression (C.4). Using the assumption that I(t) includes the values of the $1 coupon payments and the risk-free return rate r_F, and that successive information sets are nested, this substitution can be shown to give

                  $1           $1         E[P_C(t+2)|I(t)]
(C.5)  P_C(t) = -------  +  ---------  +  ----------------
                (1+r_F)     (1+r_F)^2         (1+r_F)^2

Continuing to expand the right side of (C.5) by successive substitution out for the price term, one obtains a representation for P_C(t) in terms of a discounted sum of $1 coupon payments plus a remainder term. For example, given any period L greater than t, recursing forward the far-right price term in (C.4) for L periods into the future yields the following representation for the nominal price of a consol at the beginning of period t:

                  L-t    $1            E[P_C(L)|I(t)]
(C.6)  P_C(t)  =  SUM  ---------   +   --------------    .
                  j=1 (1+r_F)^j         (1+r_F)^(L-t)

Suppose the following transversality condition holds: the far-right price term in (C.6) approaches 0 as L approaches infinity. A basic mathematical lemma states that, for any constant b with absolute value less than 1, the infinite sum of b raised to successively higher powers (starting with b^0 = 1) is 1/[1-b]. Applying this lemma to the sum in (C.6) as L approaches infinity, with b equal to 1/(1+r_F), and imposing the transversality condition, it follows that

                  +oo    $1           $1
(C.7)  P_C(t)  =  SUM ---------   =  -----     .
                  j=1 (1+r_F)^j       r_F

It is important to keep in mind that (C.7) was derived under two special assumptions: (i) EMH: For each t, the expected value of the period-t return rate (C.2) for the consol, conditional on the information set I(t), is equal to the (constant) risk-free return rate r_F; and (ii) Transversality Condition: For each t, the far-right price term in (C.6) approaches zero as L approaches infinity.

By definition, the yield to maturity for a financial asset in some period t is the particular fixed one-period interest rate R which, if used to discount the financial asset's future payment stream to the holder, would yield a present value for this payment stream that is just equal to the financial asset's period-t market value. For a consol in period t, the payment stream ($1,$1,...) simply consists of periodic $1 payments in perpetuity. Thus, the yield to maturity R for a consol bond in period t is the particular fixed one-period interest rate R that satisfies

                  +oo   $1           $1
(C.8)  P_C(t)  =  SUM  -------   =   ---   .
                  j=1 (1+R)^j         R

As previously noted, the return rate r_C(t,t+1) defined for the consol by (C.2) -- which only accounts for the $1 payment and the capital gain or loss over the holding period [t,t+1) -- can in general differ from the yield to maturity R defined by relation (C.8), which takes into account the entire (infinite) payment stream for the consol starting from time t.

The fundamental value F(t) of the consol in period t is defined to be the present value of the payment stream ($1,$1,...), where the discounting of the payment stream to present value is carried out using the risk-free return rate r_F. Thus,

                +oo    $1            $1   .
(C.9)  F(t)  =  SUM  ---------   =  ----
                j=1 (1+r_F)^j        r_F

Recall from Section B that the consol is said to exhibit a price bubble at time t if its period-t price P_C(t) deviates from its period-t fundamental value F(t). Comparing (C.6) through (C.9), it is seen that the absence of price bubbles is guaranteed for consols if two conditions hold: (i) the EMH in form (C.3); and (ii) the transversality condition (convergence to zero of the far-right price term in (C.6)).

Intuitively, it might seem that price bubbles must be absent on consols in order for the market for consols to be in equilibrium in each period t. If P_C(t) is greater than F(t) (hence R is less than r_F), why wouldn't savers switch towards the risk-free asset and away from consols in order to capture the higher yield r_F, a move that would tend to lower P_C(t) in the direction of F(t). Conversely, if P_C(t) is less than F(T) (hence R is greater than r_F), why wouldn't savers switch away from the risk-free asset and towards consols to capture the higher yield R, a move that would tend to raise P_C(t) in the direction of F(t)? Thus, F(t) appears to provide a perfectly rational way to determine the equilibrium market price of a consol in period t.

However, this intuitive reasoning implicitly assumes the existence of an infinitely-lived risk neutral "representative investor." Equilibrium requires the absence of perceived exploitable profit opportunities. Thus, given such a risk-neutral investor, equilibrium requires that the EMH in form (C.3) must hold. Moreover, the transversality condition must also hold, since otherwise the investor could take advantage of the difference between P_C(t) and F(t) by implementing a "buy and hold forever" strategy in the direction of the asset with the greater yield (either R on consols or r_F on Treasury bills).

Absent the existence of a single risk-neutral infinitely-lived investor, however, the existence of a positive price bubble on the consol is not necessarily irrational. Suppose, for example, that the market contains successive finite-lived traders with successive finite holding periods for bonds. These traders could be expecting the price P_C of the consol to accelerate over time at a rate at least equal to (1+r_F), implying that the far-right price term in (C.6) fails to converge to zero in the forward recursion. Note that these expectations of an appreciating price for consols would tend to be self-fulfilling, at least for a while, since the resulting increased demand for the consols would tend to raise the price of the consols in actuality. No single trader could take advantage of the resulting price bubble through a "buy and hold forever" strategy. Thus, the trading activities of such traders would not necessarily drive the consol price P_C(t) to its fundamental value F(t). In short, with successive finite-lived traders, the transversality condition could fail to hold even though the EMH in form (C.3) is satisfied in every period.

Recall the "dot.com bubble" in the U.S. during the late 1990s, in which many stocks for web-based companies paying zero dividends (and having negative net earnings) were nevertheless able to command a substantially positive market price on the anticipation of future price increases. For a while, until the bubble burst, lots of investors made lots of money. Were they irrational? The difficulty comes in trying to form a "rational" prediction regarding when (and if) a price bubble will burst. The issue of bubbles is currently a hot topic in economics. See, for example, the symposium on bubbles in the Journal of Economic Perspectives (Volume 4, Spring 1990).

D. Empirical Testing of the Efficient Markets Hypothesis

The EMH is given empirical content by specifying the information sets I(t) used by traders in a financial market M at each time t to determine the period-t prices of risky financial assets in accordance with relation (B.4). Fama (1970) describes a hierarchy of nested categories of information sets. As one moves down the hierarchy from the largest to the smallest set, efficiency is required with respect to ever decreasing amounts of information, and hence in an ever weaker sense. Since the categories of information sets are nested, rejection of any one type implies the rejection of all stronger forms.

An implication of weak-form efficiency (hence of any of the stronger forms of efficiency), is that asset prices from periods prior to period t will not be of any help in predicting asset prices for periods t+1 and beyond, since this past price information is already fully reflected in period-t asset prices. Literally, condition (B.4) states that the only part of the information set I(t) that is relevant for forming return rate predictions for asset A in period t is the current period-t price of asset A together with the risk-free return rate.

Given certain additional assumptions, another testable implication of the weak-form EMH (and hence all stronger forms of the EMH) is that the return rates of an asset A are not serially correlated over time. Roughly, this means that the successive return rates for asset A in successive time periods do not exhibit any systematic relationship, either positive (a tendency to move together) or negative (a tendency to move in opposite directions). For example, if the return rate for asset A happens to be higher than its average level in the current period, one cannot infer that it will be higher or lower than this average level in the following period. (See the Appendix to these notes for a more detailed discussion of this implication of weak-form EMH.)

The Empirical Bottom Line [See Sheffrin (1991, 133-137), Fama (1991)]

Throughout much of the 1970's, many economists seemed to accept the presumption that financial markets are efficient, i.e., that the EMH was satisfied, at least in its weaker forms. Nevertheless, careful reviews of the efficient market literature have always pointed out that tests of the EMH are #joint# tests of a particular asset pricing model together with the assumption that market participants make efficient use of information in determining their expectations for asset prices. Such was the faith in the EMH through the 1970's, however, that test failures generally led researchers to modify their asset pricing models rather than reject the EMH per se.

At least some researchers (e.g., Ross 1989) have concluded from the empirical literature that markets appear to be consistent with efficiency, at least in the weak-form sense. However, a growing number of researchers now take a more cautious stance. Fama (1991), for example, concludes that -- with regard to firm-specific events -- the adjustment of stock prices to new information appears to be efficient. As for other tests of efficiency, however, he reports mixed results and does not draw a conclusion. What happened?

The growing pessimism about the EMH seems to have begun in the early 1980's with the work of Shiller (1981,1989) on the "excess volatility" of financial markets. Shiller formulated statistical tests of the EMH based on the volatility of stock market prices relative to dividend volatility which he claimed were more powerful than traditional regression-based tests of the EMH. He found that stock prices diverged systematically from fundamental values, and he interpreted this finding as a violation of the EMH. This was the start of the "bubble literature" -- see the symposium on bubbles in the 1990 Journal of Economic Perspectives.

In a well-known study, Fama and French (1988) concluded that stock returns exhibit high negative correlation, implying there is a slow mean-reverting movement in stock returns. As just discussed, this violates the weak-form EMH implication that asset return rates should be serially uncorrelated.

One interpretation of mean-reversion offered by Fama and French is that expectations are still rational, but that stock returns are subject to random shocks. These shocks do not affect future dividends, hence stock returns eventually revert to their "normal" long-run value. Another interpretation, however, is that investors become inexplicably bullish or bearish. If these fads persist for a while before eventually dying out, then stock market prices will exhibit "bubbles" (divergence from fundamental value) in the short run and mean reversion in the long run; see Summers (1986).

The notion that asset prices can diverge from their fundamental values is also an implication of the "noise trader" theory developed by Black (1986). A noise trader is a market participant with incorrect information who implements trades on the basis of this information under the false belief that the information is correct. Black argued that the presence of noise traders was necessary to explain the large volume of trading activity that occurs in financial markets. Without noise traders, there would be virtually no trade in individual shares. Rational investors trading with each other would realize that any other trader willing to pay a higher price for the asset must have superior information about the asset's return. But without a large volume of trade in individual shares, how does the market determine the appropriate market prices for assets?

The presence of noise traders provides rational traders with an incentive to gather information. Sophisticated traders can bring correct information to a market and exploit the profit opportunities created by the presence of noise traders. Sophisticated traders tend to move asset prices toward fundamental values. However, the continual presence of noise traders (particularly, the continual entrance of new noise traders) makes it difficult for other traders to discern who is a noise trader and who is acting on correct information. This can give a sophisticated trader a chance to make profits from his superior information for an extended period of time.

Do sophisticated traders drive noise traders from a market? Not necessarily. For example, Shleifer and Summers (1990) argue that -- if noise traders (unwittingly) take larger risky positions on average, because of erroneous beliefs, some noise traders can still profit in the market despite their erroneous beliefs. The essential idea is that some noise traders holding large risky positions with high expected returns are lucky and manage to earn a high enough return rate on their wealth to enable them to remain in the market. In addition, the presence of noise traders actually benefits sophisticated traders who take direct advantage of them in trades, as well as other agents (brokers, dealers, etc.) who feed off noise traders indirectly by providing financial services to them.

Along similar lines, Fama (1991, p. 1607) reports that some researchers have concluded that insiders can and do profit from trading on the basis of their inside knowledge, but outsiders cannot profit from public information about this insider trading. Thus, the market may be characterized by "noisy rational expectations" (Grossman and Stiglitz, 1980) in the sense that there is private information which is not fully reflected in asset prices, but all investors nevertheless exhibit rational behavior.

All of this work on noise traders requires asymmetric information, i.e., different market participants engaging in trades with each other on the basis of different information sets.

E. References

Journal of Economic Perspectives (Spring 1990), Symposium on Bubbles.

George A. Akerlof, "The Market for Lemons," Quarterly Journal of Economics 84 (August 1970), 488-500.

W. Brian Arthur, John H. Holland, Blake LeBaron, Richard Palmer, and Paul Taylor, "Asset Pricing Under Endogenous Expectations in an Artificial Stock Market," pp. 15-44 in W. B. Arthur, S. Durlauf, and D. Lane (eds.), The Economy as an Evolving Complex System, II, Volume XXVII, Addison-Wesley, 1997.

Fischer Black, "Noise," Journal of Finance 41 (1986), 529-543.

Eugene Fama, "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance 25 (1970), 383-423.

Eugene Fama, "Efficient Capital Markets: II,"Journal of Finance 46 (December 1991), 1575-1617.

Eugene Fama and K. French, "Permanent and Temporary Components of Stock Prices," Journal of Political Economy 96 (1988), 246-273.

Mark Gertler, "Financial Structure and Aggregate Economic Activity: An Overview," Journal of Money, Credit, and Banking 20 (1988), 559-588.

Sanford Grossman and Joseph Stiglitz, "On the Impossibility of Informationally Efficient Markets," American Economic Review 70 (1980), 393-408.

Stephen LeRoy, "Efficient Capital Markets and Martingales," Journal of Economic Literature 27 (December 1989), 1583-1621.

Steven Ross, "Finance," pp. 326-329 in the New Palgrave Dictionary of Economics, J. Eatwell, M. Milgate, and P. Newman, eds., W. W. Norton and Co., New York, 1989, in four volumes.

Steven Sheffrin, The Making of Economic Policy, Blackwell, 1991.

Robert Shiller, "Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?" American Economic Review 71 (June 1981), 421-436.

Robert Shiller, Market Volatility, MIT Press, Cambridge, 1989.

Andrei Shleifer and Lawrence Summers, "The Noise Trader Approach to Finance," pp. 19-33, in Symposium on Bubbles, Journal of Economic Perspectives 4 (Spring 1990).

Lawrence Summers, "Does the stock Market Rationally Reflect Fundamental Values?" Journal of Finance 41 (1986), 591-601.

Hal Varian, Microeconomic Analysis, W. W. Norton and Co., Third Edition, 1992.

F. Appendix: Technical Remarks

As noted in Section D, above, given certain additional assumptions, another testable implication of the weak-form EMH (and hence all stronger forms of the EMH) is that the return rates of an asset A should not exhibit serial correlation.

This claim will now be shown for an asset A that distributes a dividend payment D(t,t+1) at the end of each period t = [t,t+1). The demonstration will use a famous "iterated expectation lemma" in the following special form. Let X and Y be two real-valued random variables X and Y with joint probability density function p(x,y) and with marginal probability density functions p(x) and p(y). Letting INT denote integration and INT INT denote double integration, suppose H(x,y) is any function such that the expectations

        EH(X,Y)  =   INT INT H(x,y)p(x,y)dxdy ;

        E[H(X,Y)|Y=y]  =  INT H(x,y)p(x|y)dx  ,

exist and are finite for each y. Then, using p(x,y) = p(x|y)p(y) for each x and y,

(1)     EH(X,Y)    =  INT INT H(x,y)p(x,y)dxdy

                   =  INT [ INT H(x,y)p(x|y)dx ]p(y)dy

                   =  INT E[H(X,Y)|Y=y]p(y)dy

                   =  E[ E(H(X,Y)|Y] ]  .

Now consider the covariance between the rates of return on asset A over any two consecutive time periods [t,t+1) and [t+1,t+2). Using straightforward calculations, as well as the iterated expectations lemma (1) with X = r_A(t+1,t+2) and Y = r_A(t,t+1), one obtains

(2)    Cov( r_A(t+1,t+2), r_A(t,t+1) )

         =  E[(r_A(t+1,t+2)-E[r_A(t+1,t+2)])(r_A(t,t+1)-E[r_A(t,t+1)])]

         =  E[(r_A(t+1,t+2))(r_A(t,t+1)-E[r_A(t,t+1)])]

         =  E[E[(r_A(t+1,t+2))(r_A(t,t+1)-E[r_A(t,t+1)])|r_A(t,t+1)]]

         =  E[E[(r_A(t+1,t+2))|r_A(t,t+1)](r_A(t,t+1)-E[r_A(t,t+1)])].

Suppose for each t that the risk-free interest rate r_F(t,t+1) is independent of the rate of return r_A(k,k+1) on asset A for each k less than t. Under weak-form efficiency, the time t+1 information set I(t+1) contains all current and past prices of asset A. Suppose I(t+1) also contains the dividend payment D(t,t+1). Then I(t+1) contains the return rate for asset A during period t: namely,

      r_A(t,t+1) = [p_A(t+1) + D(t,t+1) - p_A(t)]/p_A(t)  .

Making use of the iterated expectation lemma (1), as well as the EMH implication (B.1), it follows that

(3)  E[r_A(t+1,t+2)|r_A(t,t+1)] = E[E[r_A(t+1,t+2)|I(t+1)]|r_A(t,t+1)]

                                = E[r_F(t+1,t+2)|r_A(t,t+1)]

                                = E[r_F(t+1,t+2)].

Consequently, combining (2) and (3),

(4)     Cov( r_A(t+1,t+2), r_A(t,t+1) )

          = E[E[(r_A(t+1,t+2))|r_A(t,t+1)](r_A(t,t+1)-E[r_A(t,t+1)])]

          = E[E[r_F(t+1,t+2)](r_A(t,t+1)-E[r_A(t,t+1)])]

          = E[r_F(t+1,t+2)]E[r_A(t,t+1)-E[r_A(t,t+1)]

          =  0  .

In other words, the return rates for asset A are serially uncorrelated.

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